Problem: Given $ m \angle QPR = 3x + 104$, $ m \angle RPS = 8x - 72$, and $ m \angle QPS = 164$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Explanation: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {3x + 104} + {8x - 72} = {164}$ Combine like terms: $ 11x + 32 = 164$ Subtract $32$ from both sides: $ 11x = 132$ Divide both sides by $11$ to find $x$ $ x = 12$ Substitute $12$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 8({12}) - 72$ Simplify: $ {m\angle RPS = 96 - 72}$ So ${m\angle RPS = 24}$.